1. Field of the Invention
This invention relates to an improved apparatus and process for measuring linear and non-linear viscoelastic properties of viscoelastic materials.
2. Brief Description of the Prior Art
Commonly assigned U.S. Pat. No. 3,969,930 (Prevorsek et al. to Allied Chemical, 1976) discloses an apparatus and method for applying a known sinusoidal strain wave to a viscoelastic material and measuring the resulting stress in the material as a function of the applied cyclic strain. By this method, the overall total modulus of the material can be measured, and the mechanical loss, associated with the expansion-contraction cycle under non-linear viscoelastic conditions, can be determined from the area within the characteristic stress-strain hysteresis loop of the material.
Commonly assigned U.S. Pat. No. 4,056,973, (Prevorsek et al. to Allied Chemical, 1976) discloses an apparatus and method for applying a composite strain wave, comprised of a high frequency small amplitude sinusoidal strain wave superimposed onto a low frequency large amplitude fundamental sinusoidal strain wave, to a viscoelastic material, and measuring the resulting stress in the material as a function of the applied cyclic strain. By this method, the instantaneous modulus of the material can be obtained, from which conclusions can be drawn regarding; (a) the instantaneous mechanical loss as a function of applied cyclic strain, and (b) factors affecting the shape and area of the resulting stress wave or stress-strain hysteresis loop. These data provide useful information in analyzing the performance of viscoelastic structural parts in actual end uses as for example, polymeric tire cord in a rolling automobile tire, and can lead to estimates of tiretemperature profiles during service and tire rolling resistance, which in turn is ultimately and quite surprisingly related to efficiency of fuel consumption in automotive vehicles.
The above-identified references describe the analysis of resulting stress in a viscoelastic material when subjected to large amplitude fundamental sinusoidal strain waves. Under these conditions, where the applied cyclic strain is continuously varying, the material usually exhibits non-linear viscoelastic behavior.
However, the above-identified references do not disclose methods or apparatus for determining the dependence of mechanical loss upon the applied strain during applied constant rate strain for a viscoelastic material, nor for determining the instantaneous modulus during testing-to-rupture, or tensile testing, of such a material.
When a viscoelastic material is subjected to sinusoidal deformation under prestrain, the strain .gamma. varies according to the expression: EQU .gamma.(.theta.)=.gamma..sub.o +.DELTA..gamma.sin .theta. (1)
wherein .gamma.is prestrain, .DELTA..gamma. is the strain amplitude and .theta. is the angle during the cycle varying from 0.degree. to 360.degree..
When the strain amplitude is very small, the resulting stress wave is also sinusoidal but shifted on the angle scale by the phase angle difference .delta. so that stress .sigma. is represented by the expression EQU .sigma.(.theta.)=.sigma..sub.o +.DELTA..sigma.sin (.theta.+.delta.) (2)
In this case, .delta. and .DELTA..sigma. are independent of .theta.. The ratio .DELTA..sigma./.DELTA..gamma. represents the complex modulus E*, while phase angle difference is related to the mechanical loss during the cycle. Each E* and .delta. are important characteristics of viscoelastic materials. At very low strain amplitudes, the viscoelastic properties E* and .delta., are independent of the strain amplitude. Materials which under cyclic strain obey equation (2), are referred to as being linear viscoelastic.
When the strain amplitude is increased, there is observed, with all viscoelastic materials, a strain amplitude at which equation (2) no longer accurately describes the stress during the cycle. The stress wave resulting from the large amplitude sinusoidal strain wave is distorted and non-sinusoidal, and equation (2) assumes the form EQU .sigma.(.theta.)=.GAMMA..sub.o +.DELTA..sigma.(.theta.) sin [.theta.+.delta.(.theta.)]. (3)
The resulting viscoelastic properties such as dynamic modulus, mechanical loss, and the like, are then strain dependent, and the strain amplitude at which this occurs is often referred to as the limit of linear viscoelastic response. Above this limit, various materials, for example, polymeric devices (such as heart valves) and reinforced composites (such as pneumatic tires) are subjected to deformation exceeding linear response and thus exhibit non-linear viscoelastic behavior. The performance of these polymers as parts or components in actual end use under these conditions, can be correlated to the shape and area of the stress-strain hysteresis loop or stress response wave by the methods described in the above-identified references. The methods are directed to determining .DELTA..sigma.(.theta.) and .delta.(.theta.), which involve superimposing a high frequency small amplitude strain wave onto a fundamental large amplitude low frequency wave, and determining the changes in modulus as a function of strain (or angle .LAMBDA.) during the cycle and also constructing the non-linear elastic stress wave. By the term, "non-linear elastic stress wave, is meant the wave form that is calculated from the instantaneous modulus in the non-linear viscoelastic region of the stress response wave.
The instantaneous modulus is derived by superimposing a small strain amplitude (less than 1%) high frequency sinusoidal wave onto the fundamental sinusoidal wave and determining the resulting stress. By the term "strain amplitude" is meant the quantity .DELTA.L/L, where .DELTA.L is the displacement amplitude, and L is the length of the sample under pretension at the beginning of the testing procedure. The non-linear elastic stress value is then calculated from the instantaneous modulus and the fundamental strain wave. The difference in the angle between the observed stress and the calculated nonlinear elastic stress is given the symbol, .delta.', and represents the mechanical loss of a viscoelastic material under nonlinear viscoelastic conditions during an expansion-contraction cycle. When non-linear viscoelastic stress is plotted as the ordinate versus observed strain as the abscissa, a stress-strain hysteresis loop results indicating mechanical loss during the cycle, and the area of the hysteresis loop is dependent on .delta.', a measure of mechanical loss during the cycle. Plots of .delta.' as function of .theta. during applied cyclic strain, by use of the apparatus and methods of the above-identified references, show that frequently .delta.' exhibits maxima at .theta.=0.degree. and 180.degree., which suggests that with many materials .delta.' may be strainrate dependent.
Thus, an apparatus and process are needed for determining whether .delta.', and thus the shape and area of the hysteresis loop for a viscoelastic material, is affected by changes in strain rate during periodic experiments.
It would also be very desirable to measure changes in viscoelastic properties under the conditions employed in standard tensile tests. These tests are frequently carried out at constant strain rate until the specimen ruptures, but information concerning the instantaneous modulus during these processes is not readily obtainable.